The Bernstein problem for affine maximal type hypersurfaces under decaying convexity

Abstract:

We study a fourth order partial differential equation \begin{equation} u^{ij}D_{ij}w=0, \ \ w\equiv [\det D^2u]^{-\theta }, \ \ \theta \not =0\tag {\(\ast \)} \end{equation} of affine maximal type, which has attracted much attention in recent years. These models include the affine maximal equation for $\theta =\frac {N+1}{N+2}$ (see, for example, [Invent. Math. 140 (2000), pp. 399–422]) and the Abreu equation for $\theta =1$ (see, for example, [Int. J. Math. 9 (1998), pp. 641–651] or [Calc. Var. Partial Differential Equations 43 (2012), pp. 25–44] . In this paper, we will prove a Bernstein theorem of \eqref{e0.1} for all dimension $N\geq 2$ and \begin{equation*} \theta \in \Bigg (0,\frac {1}{2}-\frac {\sqrt {N-2}}{2\sqrt {N}}\Bigg )\bigcup \Bigg (\frac {1}{2}+\frac {\sqrt {N-2}}{2\sqrt {N}},+\infty \Bigg ) \end{equation*} under decaying convexity. Our result covers the affine maximal equation ($\theta =\frac {N+1}{N+2}$) for dimension $N\leq 3$ or the Abreu equation ($\theta =1$) for all dimension $N\geq 1$, which largely improves both cases by Du in [Nonlinear Anal. 187 (2019), pp. 170–179] ($N=2$ for $\theta =\frac {N+1}{N+2}$ and $N\leq 9$ for $\theta =1$).